Search results
Results from the WOW.Com Content Network
In vector calculus, the Jacobian matrix (/ dʒ ə ˈ k oʊ b i ə n /, [1] [2] [3] / dʒ ɪ-, j ɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives.
In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. These coordinates are particularly common in treating polyatomic molecules and chemical reactions, [3] and in celestial mechanics. [4] An algorithm for generating the Jacobi coordinates for N bodies may be based upon binary trees. [5]
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...
The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose ...
This is the same matrix as defines a Givens rotation, but for Jacobi rotations the choice of angle is different (very roughly half as large), since the rotation is applied on both sides simultaneously. It is not necessary to calculate the angle itself to apply the rotation. Using Kronecker delta notation, the matrix entries can be written:
Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map at each point. Therefore, in some chosen local coordinates, it is represented by the Jacobian matrix of the corresponding smooth map from R m {\displaystyle \mathbb {R} ^{m}} to R n {\displaystyle \mathbb {R} ^{n}} .
Difficult integrals may also be solved by simplifying the integral using a change of variables given by the corresponding Jacobian matrix and determinant. [1] Using the Jacobian determinant and the corresponding change of variable that it gives is the basis of coordinate systems such as polar, cylindrical, and spherical coordinate systems.
Jacobi matrix may refer to: Jacobian matrix and determinant of a smooth map between Euclidean spaces or smooth manifolds Jacobi operator (Jacobi matrix), a tridiagonal symmetric matrix appearing in the theory of orthogonal polynomials